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'''Implication''' is a [[logic]]al operation on two statements, typically represented by the variables P and Q. "P '''implies''' Q" is written symbolically as "P &rarr; Q". Equivalent statements include "'''if''' P '''then''' Q", "P is '''sufficient''' for Q", and "Q is '''necessary''' for P".

'''Implication''' is a [[logic]]al operation on two statements, typically represented by the variables P and Q. "P '''implies''' Q" is written symbolically as "P &rarr; Q". Equivalent statements include "'''if''' P '''then''' Q", "P is '''sufficient''' for Q", and "Q is '''necessary''' for P".

The statement "if P then Q" is called a '''conditional''' statement; P is known as the '''antecedent''' and Q the '''consequent'''.

The statement "if P then Q" is called a '''conditional''' statement; P is known as the '''antecedent''' and Q the '''consequent'''.

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The statement "P '''if and only if''' Q", a '''biconditional''' statement, means the same thing as "P implies Q, and Q implies P". Other equivalent meanings include "P is '''logically equivalent''' to Q", and "P is a necessary and sufficient condition for Q". The phrase "if and only if" is often abbreviated '''iff''', especially in mathematics.

==Definition==

==Definition==

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Or, rephrased to make more sense grammatically:

Or, rephrased to make more sense grammatically:

: If I end up passing the class then I must have studied hard.

: If I end up passing the class then I must have studied hard.

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==If and only if==

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The phrase "'''if and only if'''" means "is logically equivalent to", or "is a necessary and sufficient condition for". It is often abbreviated '''iff'''.

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: '''P''' if and only if '''Q'''

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means the same thing as:

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: '''P''' implies '''Q''', and '''Q''' implies '''P'''.

==Transformations of conditionals==

==Transformations of conditionals==

Revision as of 19:31, 7 December 2009

Implication is a logical operation on two statements, typically represented by the variables P and Q. "P implies Q" is written symbolically as "P → Q". Equivalent statements include "if P then Q", "P is sufficient for Q", and "Q is necessary for P".

The statement "if P then Q" is called a conditional statement; P is known as the antecedent and Q the consequent.

The statement "P if and only if Q", a biconditional statement, means the same thing as "P implies Q, and Q implies P". Other equivalent meanings include "P is logically equivalent to Q", and "P is a necessary and sufficient condition for Q". The phrase "if and only if" is often abbreviated iff, especially in mathematics.

Definition

Note that if P is true, then Q must also be true for the implication to hold (be true). However, if P is false, Q may or may not be true and the implication still holds.

To illustrate the latter fact, consider a teacher who tells her class that any student who gets 100% on the final exam will pass the class. In other words:

P: A student gets 100% on the final.

Q: That student passes the class.

P → Q: If a student gets 100% on the final, then that student passes the class.

Now consider the case in which two students do poorly on the final exam; one of them did well enough on the other exams to pass the course, but the other did not. Did the teacher lie?

No. She said nothing about students who do not get 100% on the final (i.e., the case where P is false). Unless there is a student who both got 100% on the final and did not pass the course, the teacher told the truth.

For this reason, "P → Q" can be restated as "¬(P ∧ ¬ Q)" or "not (P and not Q)", or "it is not the case that P is true and Q is false".

If the last statement were not equivalent to the original, then I might be able to not study and still pass the class — but this contradicts my original assertion that it was only by studying hard that I would pass.

Using an "unless" statement to say the same thing:

I won't pass the class unless I study hard.

This means I'll want to study hard to give myself a chance to pass, but it doesn't guarantee that I will pass; on the other hand, not studying will guarantee that I don't pass.

Symbolically, both the "only if" and "unless" versions of the statement can be written as:

¬ S → ¬ P: If I don't study hard, I won't pass the class.

where

S = I study hard.

P = I will pass the course.

Note that this is equivalent to:

P → S: If I will pass the class then I study hard.

Or, rephrased to make more sense grammatically:

If I end up passing the class then I must have studied hard.

Transformations of conditionals

Given a conditional statement "if P then Q", there are many ways of transforming the statement to other conditionals that may or may not be logically equivalent.